MATHEMATICS (MAT) Professors William Harris and Homer White (Chair); Visiting Assistant Professor Jianning Su; Visiting Lecturer Lucas Garnett The various disciplines within the Department of Mathematics, Physics and Computer Science are united by their reliance upon: methods for discovering and demonstrating patterns, and for constructing structures that exhibit, unify and illuminate these patterns; applications of these structures to model a wide variety of phenomena in mathematics and the sciences; precise language as a means to express patterns and describe structures. The Department offers majors in Mathematics and Physics. Mathematics is the art of creating, recognizing, and analyzing abstract quantitative and geometrical structures; Physics is the study of the physical universe. Students who pursue either course of study are trained to solve theoretical and practical problems and to communicate their solutions effectively. This training provides a foundation for further graduate and professional study in many fields as well as for employment in business and industry. A major in Engineering Arts is available for students who complete a specified preliminary course of study at Georgetown College that results in an engineering degree at another institution. Major (B.A. degree) Thirty-three hours required. A minimum of thirty hours in Mathematics, including MAT125, 225, 301, 310, 325, 415, and 431. The remaining nine hours in Mathematics must be chosen from courses numbered above MAT225. Allied course: CSC115. (Secondary Education Mathematics majors are required to include MAT331 and 335 in the major course selections. Total hours in Mathematics and Computer Science required: thirty-three) (B.S. degree) Thirty-nine to forty hours required. Core: twenty-four hours in Mathematics, including MAT125, 225, 301, 310, 325, 331, 332, and 415 or 431. Applied MAT Electives (six hours): two courses chosen from MAT303, 343, 345. Additional Elective (3-4 hours): one additional MAT course numbered above 301 or one of: CSC315, CSC405, PHY212, ECO304, ECO313. Allied Courses (six hours): CSC115, CSC215. 164
Minor Eighteen hours required. A minimum of eighteen hours in Mathematics, including MAT125, 225 and 301. The remaining hours must be chosen from courses numbered above MAT301. Students with strong backgrounds in mathematics will be placed at course levels commensurate with demonstrated ability. After earning twenty-four credit hours, any student who has not completed the quantitative proficiency requirement must enroll in a course that completes this requirement and in each subsequent semester must enroll in such a course until he/she completes the course with a grade of D or better or transfers in a course deemed to satisfy the requirement. For students falling under the requirement of continuous enrollment, drops during the semester can only be approved by the MPC Department Chair or the Mathematics Program Coordinator. For students with a math subscore on the ACT of less than 19 (or its equivalent), initial placement is in MAT115 Liberal Arts Mathematics. Exceptions to this placement can be made by the MPC Department Chair or the Mathematics Program Coordinator. A prerequisite must be taken before the course; a corequisite may be taken before or concurrently with a course. 107. College Algebra. (3 hours) A survey of algebraic techniques and of functions. Topics include theory of equations and inequalities, graphs, transformations of functions, inverse functions, and exponential and logarithmic functions. Can be used as preparation for MAT109. Not applicable to a major or minor in mathematics. Prerequisite: ACT Math subscore of 19 or permission of the mathematics coordinator. Fall 109. Calculus for Business and the Social Sciences. (3 hours) An introductory survey of calculus, less theoretical in nature than MAT125. Topics include derivatives of algebraic, exponential, and logarithmic functions, the definite integral, and applications to business and the social sciences. Not applicable to a major or minor in mathematics. Prerequisite: Math ACT subscore of 22 or MAT107. Students with a grade of C or higher in MAT125 (or its equivalent) may not subsequently take this course for credit. Odd Springs 111. Elementary Probability and Statistics. (3 hours) An introductory study of statistics, including such topics as numerical and graphical descriptive statistics; sampling methods and design of studies and experiments; basic probability and the distribution of sampling statistics; and inferential procedures such as confidence intervals and tests of hypothesis. This course does not count toward a major or a minor in 165
Mathematics. Prerequisite: ACT math subscore of 19 or permission of the mathematics coordinator. 115. Liberal Arts Mathematics. (3 hours) An introductory mathematics class with a focus on practical applications including but not limited to: money management, basic statistics, voting and social choice, and management science. Students with a grade of C or higher in MAT109 or MAT125 (or their equivalents) may not subsequently take this course for credit. 123. Precalculus. (3 hours) A survey of algebraic and trigonometric techniques and functions designed to prepare students for the study of calculus. Topics include a review of algebra, exponential and logarithmic functions, trigonometric functions, analytic trigonometry, and vectors. If time permits, systems of equations and conic sections will be introduced. Not applicable to a major or minor in mathematics. Prerequisite: Math ACT subscore of 22 or consent of instructor. Students with a grade of C or higher in MAT123 (or its equivalent) may not subsequently take MAT107 for credit. Fall 125. Calculus I. (3 hours) A study of the derivative, its applications, and an introduction to the integral. Topics include limits, continuity, techniques of differentiation, optimization, the Fundamental Theorem of Calculus, and indefinite integrals. Prerequisite: a C or better in MAT123 or high school precalculus and knowledge of trigonometric functions. 170. Special Topics in Mathematics (.5-3 hours) As needed 203. Mathematics for Elementary Education I. (3 hours) A detailed development of the mathematics taught in elementary school using a problem-solving approach. Topics include numeration, proportional reasoning, number theory, and, for whole numbers, fractions, and decimals, number sense and standard and non-standard algorithms. For elementary education majors only. Prerequisite: Sophomore standing. Fall 204. Mathematics for Elementary Education II. (3 hours) A continuation of MAT203. A detailed development of the mathematics taught in elementary school using a problem-solving approach. Topics include further development of the real numbering system, informal geometry, probability, and statistics. For elementary education majors only. Prerequisite: MAT203.. Spring 225. Calculus II. (3 hours) A continuation of the study of the integral and a study of multivariable differentiation. Topics include techniques of 166
integration, applications of the definite integral, vectors, partial differentiation, and Lagrange multipliers. Prerequisite: MAT125 or high school calculus. 301. Discrete Mathematics. (3 hours) An introduction to fundamental theoretical concepts of mathematics. Topics include logic, techniques of proof, elementary set theory, mathematical induction, relations and functions, counting techniques, and Boolean algebra. Corequisite: MAT225.. Fall 303. Fundamentals of Data Computing. (3 hours) This course focuses on data analysis in settings where the data is so large, dispersed or messy that machine-processing is required to gather, clean and transform it into forms suitable for analysis. We also study computer-based techniques for the analysis of such data, including machine data visualization and machine-learning. Finally we consider how the practice of reproducible research and the development of interactive web- based applications can enhance communication of the results of data analysis. Prerequisites: MAT111 or CSC115 or PSY211 or permission of the instructor. Fall 310. Linear Algebra. (3 hours) A theoretical study of systems of linear equations and vector spaces. Topics include matrix algebra, linear transformations, eigenvalues and eigenvectors, determinants, and linear programming. Prerequisites: MAT225 and 301. Spring 325. Calculus III. (3 hours) A study of infinite series, multivariable integration, and vector calculus. Topics include tests for convergence of series, power series, polar coordinates, vector-valued functions, double and triple integrals, and line integrals. Prerequisite: MAT225. Fall 327. Introduction to Numerical Methods. (3 hours) An introduction to the analysis and implementation of numerical methods. Topics include number representation and errors, locating roots of equations, interpolation, numerical differentiation, numerical integration, numerical solution of linear systems of equations, approximation by spline functions, numerical solution of differential equations, and the method of least squares. Prerequisites: CSC115 and MAT301. As needed 331. Probability Theory. (3 hours) A study of chance phenomena and probability distributions, with applications to elementary descriptive and inferential statistics. Topics include probability laws and elementary combinatorics, random variables, discrete and continuous probability distributions, the Central Limit Theorem, and basic interval estimation and hypothesis testing. Prerequisite: MAT225. Even Falls 167
332. Mathematical Statistics. (3 hours) A study of statistical methods and tests of hypotheses. Topics include estimation of parameters from both frequentist and Bayesian points of view, and linear models. Prerequisite: MAT331. Odd Springs 335. Advanced Geometry. (3 hours) A rigorous but non-axiomatic treatment of advanced geometry on the Euclidean plane, from two or more points of view. Possible viewpoints include synthetic geometry, vector geometry, and geometry using complex numbers. Usually additional topic(s) will be covered, with such topics typically being drawn from axiomatic development of elementary geometry, geometry in higher dimensions, non-euclidean geometries, and historical studies, especially geometry in non-western cultures. Prerequisites: MAT125 and 301. Odd Falls 343. Mathematical Modeling. (3 hours) An introduction to the study of modeling real-world phenomena, with an emphasis on applications to science. Topics include modeling using difference equations and differential equations, simulation, matrix modeling and Markov chains, and dimensional analysis. Prerequisite: MAT125 and CSC115. Even Springs 345. Ordinary Differential Equations. (3 hours) A study of solution methods and applications of ordinary differential equations. Topics include first order equations, second and higher order linear equations, and linear systems. Additional topics are chosen from: the Laplace transform, power series techniques, Fourier series, nonlinear systems, calculus of variations. An introduction to partial differential equations may also be included. Corequisite: MAT325. Spring 405. Complex Analysis. (3 hours) An introduction to the study of functions of one complex variable. Topics include the algebra of complex numbers, analytic functions, contour integrals, power series, the Residue Theorem, and conformal mappings. Corequisite: MAT325. As needed 413. Number Theory and Cryptology. (3 hours) A survey of topics in elementary number theory, with an emphasis on applications to cryptology. Topics include modular arithmetic, the Chinese Remainder Theorem, the Euler phi function, pseudoprimes, and various cryptosystems, including affine substitutions, the Vigenere square, and RSA. Prerequisite: MAT301. As needed 415. Abstract Algebra. (3 hours) A theoretical development of basic algebraic structures, with an emphasis on group theory. Topics include cyclic groups, Lagrange s Theorem, quotient groups, and homomorphisms. Rings, integral domains, and fields are introduced. Prerequisites: 168
MAT225 and 301. Even Falls 431. Real Analysis I. (3 hours) A theoretical development of the elements of calculus. Topics include sequences, continuity, derivatives, and integrals of single-variable functions. Prerequisites: MAT225 and 301. Odd Falls 432. Real Analysis II. (3 hours) A continuation of MAT431. Topics include integration theory, infinite series, and series and sequences of functions. Prerequisite: MAT431. Even Springs 440. Independent Study. (1-3 hours) As needed 460. Internship. (1-3 hours) Students may receive graduation credit for internships with appropriate disciplinary content that meet the facultyapproved criteria for academic internships. Such experiences include a significant reflective component and must be supervised by a full-time member of the Georgetown College faculty. Prerequisites: consent of the supervising instructor. As needed 470. Advanced Topics. (0.5-3 hours) As needed 169